For pure \(\mathcal {S}\mathcal {E}\)  states in the purely decohering scenario, Eq. (REF )
and Eq. (REF ), \(c\)  can be any value between 0 and
1 within the generalized overlap contribution, \(\mathrm {tr}\left[\rho _{\left.k\right|1}^{c}\rho _{\left.k\right|2}^{1-c}\right]\) ,
and it will give the exact overlap \(\left|\langle \psi _{\left.k\right|1}|\psi _{\left.k\right|2}\rangle \right|^{2}=\left|\gamma _{k}\right|^{2}\) 
(which is also the decoherence factor \(\gamma _{k}\)  squared for this
case of pure states). Touil et al. [1]}
consider the homogeneous case where \(\gamma _{k}=\gamma \)  for all \(k\) ,
which I will also consider (see Refs. [2]}, [3]}
for inhomogeneous results).
